Physical interpretation of initial conditions
for fractional differential equations
with Riemann-Liouville fractional derivatives1
Nicole Heymans(a) and Igor Podlubny(b)
arXiv:math-ph/0512028v1 9 Dec 2005
(a)
Physique des Matériaux de Synthèse,
Université Libre de Bruxelles,
CP259, boulevard du Triomphe, 1050 Bruxelles, Belgium
e-mail: nheymans@ulb.ac.be
(b)
B.E.R.G. Faculty, Technical University of Kosice,
B. Nemcovej 3, 04200 Kosice, Slovak Republic
e-mail: igor.podlubny@tuke.sk
September 1, 2005
Abstract
On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives,
and that it is possible to obtain initial values for such initial conditions
by appropriate measurements or observations.
1
Introduction
Many physical phenomena lead to their description in terms of non integer
order differential equations. Formulations of non integer order derivatives,
generally called fractional derivatives, fall into two main classes: RiemannLiouville derivatives and Grünwald-Letnikov derivatives, on one hand, defined as (Podlubny 1999, Samko et al. 1993)
α
0 Dt f (t)
1
=
Γ(n − α)
d
dt
1
!nZt
0
f (τ ) dτ
,
(t − τ )α−n+1
(1)
Pre-print of the article published in Rheologica Acta, Online First, November 29, 2005).
The original publication is available at www.springerlink.com
Original article DOI: 10.1007/s00397-005-0043-5.
1
or the Caputo derivative on the other, defined as (Caputo and Mainardi
1971)
Zt
1
f (n) (τ ) dτ
C α
D
f
(t)
=
,
(2)
0
t
Γ(n − α) (t − τ )α−n+1
0
where n − 1 ≤ α < n.
In this article we deal only with the Riemann-Liouville fractional derivatives. Fractional differential equations in terms of the Riemann-Liouville
derivatives require initial conditions expressed in terms of initial values of
fractional derivatives of the unknown function (Podlubny 1999, Samko et
al. 1993), like, for example, in the following initial value problem (where
n − 1 < α < n):
α
0 Dt f (t)
h
+ af (t) = h(t);
α−k
f (t)
0 Dt
i
t→0
= bk ,
(t > 0)
(k = 1, 2, . . . , n).
(3)
(4)
On the contrary, initial conditions for the Caputo derivatives are expressed in terms of initial values of integer order derivatives. It is known
that for zero initial conditions the Riemann-Liouville, Grünwald-Letnikov
and Caputo fractional derivatives coincide (Podlubny 1999). This allows a
numerical solution of initial value problems for differential equations of non
integer order independently of the chosen definition of the fractional derivative. For this reason, many authors either resort to Caputo derivatives, or
use the Riemann-Liouville derivatives but avoid the problem of initial values
of fractional derivatives by treating only the case of zero initial conditions.
It is frequently stated that the physical meaning of initial conditions
expressed in terms of fractional derivatives is unclear or even non existent.
The old and ubiquitous requirement for physical interpretation of such initial
conditions was most clearly formulated recently by Diethelm et al. (2005):
“A typical feature of differential equations (both classical and
fractional) is the need to specify additional conditions in order
to produce a unique solution. For the case of Caputo FDEs,
these additional conditions are just the static initial conditions
. . . , which are akin to those of classical ODEs, and are therefore familiar to us. In contrast, for Riemann-Liouville FDEs,
these additional conditions constitute certain fractional derivatives (and/or integrals) of the unknown solution at the initial
point x = 0 . . . , which are functions of x. These initial conditions are not physical ; furthermore, it is not clear how such
quantities are to be measured from experiment, say, so that they
can be appropriately assigned in an analysis.”
2
(Emphasis is ours). This quotation highlights the utmost importance of
the interpretation of initial conditions in terms of fractional derivatives for
further applications in various fields of science. The physical and geometric
interpretations of operations of fractional integration and differentiation were
suggested recently by Podlubny (2002). However, the problem of interpretation of initial conditions still remained open.
In this paper we shall show that initial conditions for fractional differential
equations with Riemann-Liouville derivatives expressed in terms of fractional
derivatives have physical meaning, and that the corresponding quantities can
be obtained from measurements. We shall also demonstrate that in many
instances of practical significance zero initial conditions, which are used so
frequently in practice, appear in a natural way.
2
Number of initial conditions, past history
and memory
When a physical process can be described in terms of a differential equation
of integer order n, it is well known that n conditions are required to solve the
system. In this paper only initial conditions are considered. It is also known
(Podlubny 1999, Samko et al. 1993) that fractional differential equations of
order α require α∗ initial conditions, where α∗ is the lowest integer greater
than α. This means that if α < 1 as is the case in viscoelasticity when inertial
effects are negligible, a single initial condition is sufficient. However, one of
the reasons for the success encountered in describing viscoelasticity by means
of differential equations of non integer order is their ability to describe real
behaviour, including memory effects such as are observed in polymers, using
only a restricted number of material parameters. Such memory effects may
continue to affect the material response long after the cause has disappeared,
as observed in stress relaxation after a non monotonous loading programme
(Heymans and Kitagawa 2004). In such a case a single initial condition would
appear insufficient to predict material response.
Here we shall consider only the response of a system starting at t = 0
from a state of absolute rest. As a further simplification, we shall consider
only response to ideal loading programs, such as step or impulse response.
The effects of a finite loading time, of the details of the loading program, and
of past history will be accounted for separately in a sequel.
It has been shown (Beris and Edwards 1993) that thermodynamically
valid constitutive equations for viscoelasticity are completely equivalent to
analog models containing only elements (springs and dashpots) with positive
3
coefficients. A suitable hierarchical arrangement of springs and dashpots
gives rise to spring-pot behaviour (described below), either exactly at all
timescales for an infinite tree (Heymans and Bauwens 1994), or in the longterm (or low-frequency) limit for an infinite ladder or infinite Sierpinski gasket
(Heymans and Bauwens 1994, Schiessel and Blumen 1993, 1995). In the latter
case short-term behaviour is similar to a Maxwell model with one element
replaced by a spring-pot. Therefore the equivalence demonstrated by Beris
and Edwards can be generalized to models including spring-pots (Heymans
1996), hence discussion here will be limited to such models.
3
Spring-pot model
We shall start with a spring-pot alone, which is a linear viscoelastic element
whose behaviour is intermediate between that of an elastic element (spring)
and a viscous element (dashpot). The term “spring-pot” was introduced
by Koeller (1984), although the concept of an element with intermediate
properties had been introduced some time earlier. The constitutive equation
of a spring-pot is:
1
−α
(5)
0 Dt σ(t)
K
where σ is stress, ǫ is strain and K is the model constant. The spring-pot is
the viscoelastic version of Westerlund’s “simplest model” (Westerlund 2002).
If α = 0 the element is linear elastic (Hookean spring) whereas if α = 1 it is
purely viscous (Newtonian dashpot). Insight can be gained from the response
of a spring-pot in a few simple cases, using the general relationship (Podlubny
1999, Samko et al. 1993)
σ(t) = K 0 Dtα ǫ(t)
p
α
0 Dt (at )
3.1
or
=a
ǫ(t) =
Γ(1 + p) p−α
t .
Γ(1 + p − α)
(6)
Creep or general finite load
In the case of creep, a stress step σ0 is applied at initial time t = 0. The
strain response is hence ǫ(t) = (σ0 /KΓ(1 + α))tα . The initial value of the
strain vanishes, i.e. there is no instantaneous (elastic) strain, only an anelastic (retarded) response. However, the first ordinary derivative of strain is
unbounded, so that a finite though undefined strain can be reached in an
arbitrarily small time interval.
The change of ǫ(t) is described by the fractional differential equation
σ0
α
(7)
0 Dt ǫ(t) =
K
4
In accordance with the theory of fractional differential equations in terms
of Riemann-Liouville derivatives, an initial condition involving 0 Dtα−1 ǫ(t) is
required. This condition can be found by taking the first-order integral of
the constitutive equation as
h
h
α−1
ǫ(t)
0 Dt
i
t→0
h
=
−1
0 Dt (σ0 /K)
i
t→0
.
In the
i case under consideration stress is finite at all times, hence
−1
D
σ
= 0, which leads to zero initial condition for 0 Dtα−1 ǫ(t), namely
0 t
0
t→0
h
α−1
ǫ(t)
0 Dt
i
t→0
= 0.
(8)
The same considerations apply to a general finite load σ(t). In the latter
case the equation to be solved is
α
0 Dt ǫ(t)
=
σ(t)
,
K
(9)
and the initial condition to be attached to this equation is the zero initial
condition (8).
3.2
Stress relaxation or general deformation
The stress response to a strain step ǫ0 is σ(t) = (ǫ0 K/Γ(1 − α))t−α . The
initial stress is unbounded reflecting the fact that a spring-pot (just like a
dashpot) cannot respond immediately to a bounded stress: it has an infinite
initial modulus or a vanishing initial compliance. However, relaxation to a
finite though undefined stress occurs in an arbitrarily small time interval.
The change of σ(t) is described by the fractional differential equation
−α
0 Dt σ(t)
= Kǫ0 .
(10)
From the known value of ǫ0 we can obtain the initial value (as t approaches
zero) of 0 Dt−α σ(t). Clearly, if 0 Dt−α σ(t) is to be finite although it is defined
over a vanishingly small time interval, σ(t) must be unbounded. On the
contrary, the initial value of 0 Dt−α σ(t) is well defined and finite, and that of
−α−1
σ(t) is zero. Thus, contrary to the idea expressed by some authors
0 Dt
(e.g., Glöckle and Nonnenmacher 1991), initial value problems expressed in
terms of fractional integrals are not better posed than those expressed in
terms of fractional derivatives.
If strain increases linearly with time, stress increases as t1−α . Stress is
bounded, but the initial values of its integer order derivatives are unbounded.
5
The known strain rate allows us to define the initial value of 0 Dt1−α σ(t). In
fact, in this case, zero initial conditions are found both for 0 Dt−α σ(t) and
−α−1
σ(t).
0 Dt
For any general finite strain ǫ(t), following the same reasoning, again zero
initial conditions are found.
In all three examples given here, initial conditions expressed in terms
of fractional derivatives or integrals arise naturally when taking measurable
quantities into account.
3.3
Impulse response
The impulse response is seldom used in viscoelasticity except as a mathematical convenience, because it is even more problematic to apply a homogeneous
impulse of stress or strain on a sample than it is to apply a step. However,
we shall investigate the impulse response following the same reasoning as for
the step response above.
Consider an impulse of stress defined as Bδ(t) applied to the spring-pot
at time t = 0. After that, the stress remains zero. The strain response is
ǫ(t) = (B/KΓ(α))tα−1 . The initial stress singularity gives rise to a lowerorder strain singularity, since a spring-pot cannot deform immediately.
The strain ǫ(t) for t > 0 is the solution to the fractional differential
equation
α
(11)
0 Dt ǫ(t) = 0.
In accordance with the theory of fractional differential hequations with
i
Riemann-Liouville derivatives, an initial condition involving 0 Dtα−1 ǫ(t)
t→0
is required. This can be found through integration of the constitutive equation, as
h
i
h
i
α−1
ǫ(t)
= 0 Dt−1 σ(t)/K
= B/K,
0 Dt
t→0
t→0
which gives the following initial condition to equation (11):
h
α−1
ǫ(t)
0 Dt
i
t→0
= B/K.
(12)
In this problem in terms of Riemann-Liouville derivatives B is the initial
impulse of stress σ(t), ǫ(t) is the strain after application of this impulse, and
the known impulse of stress yields a non-zero initial condition (12) involving
a fractional derivative of strain. This fractional derivative is non zero, well
defined, and bounded. Note that both strain and its integer-order derivatives
are unbounded, and its first order integral is zero, so that a meaningful
initial condition expressing the loading conditions cannot be obtained using
integral-order derivatives.
6
The physically unrealistic stress response to a prescribed strain impulse
will not be considered here. In fact, the analytical solution has a strong
t−(1+α) divergence, reflecting the fact that a strain impulse cannot be applied
to a spring-pot.
4
The key: look for inseparable twins
Now, after introducing the above simple example, let us formulate our general approach to interpretation of initial conditions involving the RiemannLiouville fractional derivatives.
In a general case, when we consider some fractional differential equation
for, say, U(t), we have to consider also some function V (t), for which some
dual relation exists between U(t) and V (t). For example, in viscoelasticity
we have to consider the pair of stress σ(t) and strain ǫ(t); in electrical circuits
the pair of current i(t) and voltage v(t); in heat conduction the pair of the
temperature difference T (t) and the heat flux q(t); etc. Functions U(t) and
V (t) are normally related by some basic physical law for the particular field
of science. In each scientific field there are such pairs of functions like the
aforementioned, which are as inseparable as Siamese twins: the left-hand
side of the initial condition involves one of them, whereas the evaluation of
the right-hand side is related to the other.
This concept is not restricted to the spring-pot treated above, but is
further applied in the subsequent sections to more elaborate models of viscoelastic behaviour. Indeed, a spring-pot is a particularly crude model, which
has several unrealistic and unphysical characteristics. As pointed out above,
it has a vanishing initial compliance or an infinite initial modulus. Viscoelastic solids, on the contrary, have a well-defined instantaneous modulus. (Note
that an unbounded initial modulus is no more of a problem when describing
a viscoelastic fluid than it is when describing Newtonian viscosity: if a step
strain is applied to a dashpot, the initial stress is also unbounded). At long
times there is no limit to anelastic strain of a spring-pot: creep continues
indefinitely. Also, stress relaxes to vanishingly small values. The increase of
stress in constant strain-rate conditions means that if attempting to describe
a viscoelastic fluid, steady state flow is never attained. When describing a
viscoelastic solid, again we find an unbounded modulus at t = 0. In spite
of these limitations, the spring-pot can give an approximate description of
polymer viscoelasticity in the intermediate time range. Several slightly more
elaborate models, which alleviate some oversimplifications of a single springpot, will be investigated below.
7
5
The fractional order Voigt model
The fractional Voigt model (a spring and a spring-pot in parallel) is nowadays generally understood as a long-term approximation to the fractional
Zener model and it might seem irrelevant to express concern over initial
value problems for the Voigt model. However, as the purpose of this note
is mainly to examine how initial conditions endowed with physical meaning
may be expressed in systems whose constitutive equations contain fractional
derivatives, we shall continue to examine the fractional Voigt model.
The constitutive equation of this model is
σ(t) = Eǫ(t) + K 0 Dtα ǫ(t).
(13)
The Voigt element or associations thereof are considered in viscoelasticity
modelling to be appropriate to obtain the strain response to a prescribed
stress program, so we shall investigate only such cases here.
Assume a stress impulse Bδ(t) is applied to a Voigt element at time t = 0.
Then the fractional order equation we need to solve for ǫ(t) (t > 0) is
Eǫ(t) + K 0 Dtα ǫ(t) = 0.
(14)
In agreement with the theory of fractional differential equations in terms
of Riemann-Liouville derivatives, we need an initial condition, which will
involve the value of 0 Dtα−1 ǫ(t) for t → 0. This condition can be obtained by
integration of the constitutive equation as
h
E 0 Dt−1 ǫ(t) + K 0 Dtα−1 ǫ(t) = 0 Dt−1 σ(t)
i
t→0
.
(15)
The limit of the right hand side is the magnitude B of the stress impulse.
On physical grounds, the spring-pot cannot deform instantaneously under a
finite stress, and, as is the case for a spring-pot alone, any singularity of ǫ(t)
must be weaker than that of the stress impulse, thus
h
−1
0 Dt ǫ(t)
i
t→0
= 0.
This can also be found fromh examination
of the behaviour of
i
h the left hand
i
−1
side of the relationship (15): if 0 Dt ǫ(t)
is non zero, then 0 Dtα−1 ǫ(t)
t→0
t→0
is unbounded and equation (15) cannot be fulfilled. Hence the initial condition finally takes on the form of
h
K 0 Dtα−1 ǫ(t)
i
t→0
= B.
(16)
This condition expresses the initial value of the fractional derivative of
strain, 0 Dtα−1 ǫ(t), in terms of the stress impulse. We see that the initial
8
condition is obtained expressing a fractional derivative of the unknown strain
in terms of a measurable, physically meaningful value of its “inseparable
Siamese twin”– the stress. The obtained initial condition is, in fact, identical
to the initial condition of the spring-pot alone. This reflects the known fact
that the spring in the Voigt model only affects long-term behaviour.
Now let us consider creep, i. e. the response to a stress step σ0 applied
at t = 0. The equation to be solved for the strain ǫ(t) is
Eǫ(t) + K 0 Dtα ǫ(t) = σ0 ,
(17)
and the initial condition for (17) can be found from
h
E 0 Dt−1 ǫ(t) + K 0 Dtα−1 ǫ(t) = 0 Dt−1 σ(t)
i
t→0
,
where the limit of the right hand side is zero. A bounded stress can produce
only a bounded strain, so the limit of the first-order ordinary integral of
strain in the left hand side is also zero. Thus the initial condition has the
form:
h
i
α−1
D
ǫ(t)
= 0.
(18)
0 t
t→0
Once more, knowledge of a measurable quantity (σ0 ) leads to an initial
condition expressed in terms of a fractional order derivative of the unknown
(ǫ(t)), its inseparable Siamese twin.
The case of a general finite stress program is similar to that of creep. The
equation to be solved is now
Eǫ(t) + K 0 Dtα ǫ(t) = σ(t),
(19)
and the initial condition is identical to the condition (18) obtained in creep.
6
The fractional order Maxwell model
To keep to a simple model while describing realistic behaviour for a viscoelastic solid, a spring expressing instantaneous elasticity must be associated in
series with the spring-pot. This eliminates the unbounded initial stress in
description of relaxation. The Maxwell element or associations thereof are
considered in viscoelasticity modelling to be appropriate to obtain the stress
response to a prescribed strain program, so we shall investigate only such
cases here.
The constitutive equation of the Maxwell model is
ǫ(t) =
1
1
σ(t) + ( 0 Dt−α σ(t)),
E
K
9
or
1
1
α
σ(t) = 0 Dtα ǫ(t).
(20)
0 Dt σ(t) +
E
K
In stress relaxation, a step strain ǫ0 is applied at t = 0. Then the equation
to be solved is
1
ǫ0 t−α
1
α
σ(t) =
.
(21)
0 Dt σ(t) +
E
K
Γ(1 − α)
h
i
An initial condition is required, involving the value of 0 Dtα−1 σ(t)
.
t→0
Integrating the constitutive equation (20) and considering the limit as t approaches zero, we have
1
1
α−1
σ(t) + 0 Dt−1 σ(t) = 0 Dtα−1 ǫ(t)
0 Dt
E
K
.
(22)
t→0
Since strain remains bounded during loading, and α < 1, the right hand
side inside brackets is bounded and vanishes when t → 0. Since the left hand
side is a linear combination of positive functions with positive coefficients,
it can only vanish if each term vanishes. This means that stress remains
bounded during loading, and hence that we obtain the following initial condition:
h
i
α−1
D
σ(t)
= 0.
(23)
0 t
t→0
Here again we observe that the initial condition on the unknown stress
arises naturally from its Siamese twin, the known strain.
Now we shall consider the strain impulse response. A strain impulse of
magnitude Aδ(t) is applied at time t=0. Thereafter, the equation to be
solved is
1
1
σ(t) + 0 Dt−α σ(t) = 0.
(24)
E
K
The required initial condition is obtained as above by integrating the
constitutive equation (20) and considering the limit as t approaches zero:
1
1
−1
−α−1
σ(t) = 0 Dt−1 ǫ(t)
.
(25)
0 Dt σ(t) +
0 Dt
E
K
t→0
The limit of the right hand side of (25) is A. Hence the limit of the left
hand side must also be bounded. This means that the limit of the first-order
integral of stress must be bounded, and the α + 1 integral must vanish, and
the initial condition is finally
1
−1
0 Dt σ(t)
E
10
= A.
t→0
(26)
The singularity in the stress response to a strain impulse is now of the
same order as that of the strain impulse itself: adding a spring in series with
the spring-pot has weakened the singularity.
The strain response to a stress impulse is identical to that of a spring-pot
alone since the impulse response of the spring vanishes.
7
The fractional order Zener model
Among the fractional order models of viscoelasticity considered in this article,
the most general is the Zener model. Its constitutive equation is
σ(t) + ν 0 Dtα σ(t) = λ ǫ(t) + µ 0 Dtα ǫ(t),
(27)
where λ = E∞ is the long-term modulus, µ = K(E0 − E∞ )/E0 , ν = µ/E0
and E0 is the instantaneous modulus.
Let us first investigate the response to a stress impulse Bδ(t) applied to
the Zener element at time t = 0. Then the fractional differential equation we
need to solve for ǫ(t) (t > 0) is:
λǫ(t) + µ 0 Dtα ǫ(t) = 0.
(28)
In accordance with the theory of fractional differential equations, we need
an initial condition involving the initial value of 0 Dtα−1 ǫ(t). Integration of
the constitutive equation gives:
−1
0 Dt σ(t)
+ ν 0 Dtα−1 σ(t) = λ 0 Dt−1 ǫ(t) + µ 0Dtα−1 ǫ(t).
(29)
The initial condition can be found by considering equation (29) as t → 0:
h
−1
α−1
σ(t) = λ 0 Dt−1 ǫ(t) + µ 0 Dtα−1 ǫ(t)
0 Dt σ(t) + ν 0 Dt
i
t→0
.
(30)
Using considerations similar to those in case of the Voigt model under
stress impulse, we obtain the initial condition in the form:
h
µ 0 Dtα−1 ǫ(t)
i
t↔0
=B
(31)
As in case of the Voigt model, this condition gives the initial value of the
fractional derivative of unknown strain, 0 Dtα−1 ǫ(t), in terms of its “inseparable twin” – the stress.
The right and left hand sides of equations (28) and (30) are formally
identical, hence following the same reasoning as above the response to a
11
strain impulse Aδ(t) applied to the Zener element at time t = 0 will be the
solution to the equation
σ(t) + ν 0 Dtα σ(t) = 0
(32)
h
(33)
with the initial condition
ν 0 Dtα−1 σ(t)
i
t↔0
=A
This formal equivalence between response to a stress or strain impulse
reflects the well known fact that the Zener model is the simplest model
capable of describing response to a stress or strain program equally well.
In case of creep, i.e. a step-stress σ(t) = σ0 for σ > 0, we have the
equation for ǫ(t):
λǫ(t) + µ 0 Dtα ǫ(t) = σ0 + ν σ0
t−α
.
Γ(1 − α)
(34)
The initial condition can also be found by considering equation (29) as
t → 0.
Following a similar reasoning to that given above for the Maxwell model
in stress relaxation, we find a zero initial condition to accompany equation
(34):
h
i
α−1
ǫ(t)
= 0.
(35)
0 Dt
t→0
This initial condition in terms of fractional derivative of ǫ(t) appeared
from consideration of its “inseparable twin” σ(t).
Similarly, for stress relaxation, ǫ(t) = ǫ0 , we obtain the equation for σ(t):
σ(t) + ν
α
0 Dt σ(t)
t−α
= λǫ0 + µ ǫ0
.
Γ(1 − α)
(36)
The initial condition for the unknown stress σ(t),
h
α−1
σ(t)
0 Dt
i
t→0
= 0,
(37)
appears naturally from consideration of the initial value of strain.
Let us now consider the case of general load σ(t) = σ∗ (t). The equation
to be solved for ǫ(t) is
λǫ(t) + µ 0 Dtα ǫ(t) = σ∗ (t) + ν 0 Dtα σ∗ (t)
(38)
The corresponding initial condition can be obtained using the following
procedure. Consider some small t = a. Starting at t = 0, stress σ(t) must be
12
recorded until t = a, and based on the recorded values the left hand side of
the relationship (29) must be evaluated. The obtained quantity provides an
approximation of the initial value for the expression in the right hand side
of (29).
In some cases it is possible to find the limit of such approximation as
a → 0. For example, for a physically realisable continuous load σ∗ (t) we
obtain a zero initial condition in the form:
h
α−1
ǫ(t)
0 Dt
i
t→0
= 0.
(39)
It is worth mentioning that this procedure amounts, in fact, to the same
as measuring the initial value of, for example, the first derivative in the case
of classical differential equations of integer order. From the examples given
above, it can be seen that for any physically realistic model, zero initial
conditions will be found for a continuous loading program or even in the case
of a step discontinuity. Non-zero conditions will only be found in the case of
an impulse.
8
Conclusion
In this note we demonstrated on a series of examples that it is possible to attribute physical meaning to initial conditions expressed in terms of RiemannLiouville fractional derivatives.
To summarize, expressing initial conditions in terms of fractional derivatives of a function U(t) is not a problem, because it does not require a direct
experimental evaluation of these fractional derivatives. Instead, one should
consider its “inseparable twin” V (t) related to U(t) via a basic physical law,
and measure (or consider) its initial values.
It is worth noting that the only case where non zero initial conditions
appeared in our considerations, is the case of impulse response. In other
cases (including the Zener model under physically realisable load program),
the initial conditions are zero, and in such cases the use of the RiemannLiouville derivatives, the Grünwald-Letnikov derivatives, and the Caputo
derivatives is equivalent.
References
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13
[2] Caputo M, Mainardi F. (1971) Linear models of dissipation in anelastic
solids. Riv. Nuovo Cimento (Ser. II) 1 :161-198
[3] Diethelm K, Ford NJ, Freed AD, Luchko Yu (2005) Algorithms for the
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